grmhd::Solutions::BondiMichel Class Reference

Bondi-Michel accretion [137] with superposed magnetic field in Schwarzschild spacetime in Cartesian Kerr-Schild coordinates. More...

#include <BondiMichel.hpp>

Classes
struct	IntermediateVars
struct	MagFieldStrength
	The strength of the radial magnetic field. More...
struct	Mass
	The mass of the black hole. More...
struct	PolytropicExponent
	The polytropic exponent for the polytropic fluid. More...
struct	SonicDensity
	The rest mass density of the fluid at the sonic radius. More...
struct	SonicRadius
	The radius at which the fluid becomes supersonic. More...

Public Types
using	equation_of_state_type = EquationsOfState::PolytropicFluid< true >
using	options = implementation defined
Public Types inherited from grmhd::AnalyticSolution
template<typename DataType >
using	tags = implementation defined

Public Member Functions
	BondiMichel (const BondiMichel &)=default
BondiMichel &	operator= (const BondiMichel &)=default
	BondiMichel (BondiMichel &&)=default
BondiMichel &	operator= (BondiMichel &&)=default
	BondiMichel (double mass, double sonic_radius, double sonic_density, double polytropic_exponent, double mag_field_strength)
auto	get_clone () const -> std::unique_ptr< evolution::initial_data::InitialData > override
template<typename DataType , typename... Tags>
tuples::TaggedTuple< Tags... >	variables (const tnsr::I< DataType, 3 > &x, const double, tmpl::list< Tags... >) const
	Retrieve a collection of hydro variables at (x, t)
template<typename DataType , typename Tag >
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataType, 3 > &x, const double, tmpl::list< Tag >) const
void	pup (PUP::er &) override
const EquationsOfState::PolytropicFluid< true > &	equation_of_state () const
virtual auto	get_clone () const -> std::unique_ptr< InitialData >=0

Static Public Attributes
static constexpr Options::String	help
Static Public Attributes inherited from grmhd::AnalyticSolution
static constexpr size_t	volume_dim = 3_st

Protected Member Functions
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::RestMassDensity< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::RestMassDensity< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::ElectronFraction< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::ElectronFraction< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificInternalEnergy< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::SpecificInternalEnergy< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Pressure< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::Pressure< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::Temperature< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::Temperature< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpatialVelocity< DataType, 3 > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::SpatialVelocity< DataType, 3 > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::MagneticField< DataType, 3 > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::MagneticField< DataType, 3 > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::DivergenceCleaningField< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::DivergenceCleaningField< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::LorentzFactor< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::LorentzFactor< DataType > >
template<typename DataType >
auto	variables (const tnsr::I< DataType, 3 > &x, tmpl::list< hydro::Tags::SpecificEnthalpy< DataType > >, const IntermediateVars< DataType > &vars) const -> tuples::TaggedTuple< hydro::Tags::SpecificEnthalpy< DataType > >
template<typename DataType , typename Tag , Requires< not tmpl::list_contains_v< tmpl::push_back< hydro::grmhd_tags< DataType >, hydro::Tags::SpecificEnthalpy< DataType > >, Tag > > = nullptr>
tuples::TaggedTuple< Tag >	variables (const tnsr::I< DataType, 3 > &, tmpl::list< Tag >, const IntermediateVars< DataType > &vars) const

Protected Attributes
double	mass_ = std::numeric_limits<double>::signaling_NaN()
double	sonic_radius_ = std::numeric_limits<double>::signaling_NaN()
double	sonic_density_ = std::numeric_limits<double>::signaling_NaN()
double	polytropic_exponent_ = std::numeric_limits<double>::signaling_NaN()
double	mag_field_strength_ = std::numeric_limits<double>::signaling_NaN()
double	sonic_fluid_speed_squared_
double	sonic_sound_speed_squared_
double	polytropic_constant_ = std::numeric_limits<double>::signaling_NaN()
double	mass_accretion_rate_over_four_pi_
double	bernoulli_constant_squared_minus_one_
double	rest_mass_density_at_infinity_
EquationsOfState::PolytropicFluid< true >	equation_of_state_ {}
gr::Solutions::KerrSchild	background_spacetime_ {}

Friends
bool	operator== (const BondiMichel &lhs, const BondiMichel &rhs)

Detailed Description

Bondi-Michel accretion [137] with superposed magnetic field in Schwarzschild spacetime in Cartesian Kerr-Schild coordinates.

An analytic solution to the 3-D GRMHD system. The user specifies the sonic radius $r_c$ and sonic rest mass density ${\rho }_c$, which are the radius and rest mass density at the sonic point, the radius at which the fluid's Eulerian velocity as seen by a distant observer overtakes the local sound speed $c_{s,c}$. With a specified polytropic exponent $\gamma$, these quantities can be related to the sound speed at infinity $c_{s,\mathrm{\infty }}$ using the following relations:

$$\begin{array}{rl}{c}_{s,c}^2& =\frac{M}{2r_c-3M}\\ c_{s,\mathrm{\infty }}^2& =\gamma -1+(c_{s,c}^2-\gamma +1)\sqrt{1+3c_{s,c}^2}\end{array}$$

In the case of the interstellar medium, the sound speed is $\approx {10}^{-4}$, which results in a sonic radius of $\approx {10}^{8}M$ for $\gamma \ne 5/3$ [171].

The density is found via root-finding, through the Bernoulli equation. As one approaches the sonic radius, a second root makes an appearance and one must take care to bracket the correct root. This is done by using the upper bound $\frac{\dot{M}}{4\pi }\sqrt{\frac{2}{M{r}^3}}$.

Additionally specified by the user are the polytropic exponent $\gamma$, and the strength parameter of the magnetic field $B_0$. In Cartesian Kerr-Schild coordinates $(x,y,z)$, where $r=\sqrt{x^2+y^2+z^2}$, the superposed magnetic field is [67]

$$\begin{array}{r}{B}^i(\overrightarrow{x},t)=\frac{B_0{M}^2}{r^3\sqrt{\gamma }}{x}^i=\frac{B_0{M}^2}{r^3\sqrt{1+2M/r}}{x}^i.\end{array}$$

The accretion rate is

$$\begin{array}{r}\dot{M}=4\pi r^2\rho u^r,\end{array}$$

and at the sonic radius

$$\begin{array}{r}{\dot{M}}_c=\sqrt{8}\pi \sqrt{M}{r}_c^{3/2}{\rho }_c.\end{array}$$

The polytropic constant is given by

$$\begin{array}{r}K=\frac{1}{\gamma {\rho }_c^{\gamma -1}}\left[\frac{M(\gamma -1)}{(2r_c-3M)(\gamma -1)-M}\right].\end{array}$$

The density as a function of the sound speed is

$$\begin{array}{r}{\rho }^{\gamma -1}=\frac{(\gamma -1)c_s^2}{\gamma K(\gamma -1-c_s^2)}.\end{array}$$

Horizon quantities, \f$\gamma\ne5/3\f$

The density at the horizon is given by:

$$\begin{array}{r}{\rho }_h\simeq \frac{1}{16}{\left(\frac{5-3\gamma }{2}\right)}^{(3\gamma -5)/[2(\gamma -1)]}\frac{{\rho }_{\mathrm{\infty }}}{c_{s,\mathrm{\infty }}^3}.\end{array}$$

Using the Lorentz invariance of $b^2$ we evaluate:

$$\begin{array}{r}{b}^2=\frac{B^2}{W^2}+(B^i{v}_i{)}^2=B^r{B}^r(1-{\gamma }_{rr}{v}^r{v}^r)+B^r{B}^r{v}^r{v}^r=B^r{B}^r=\frac{B_0^2{M}^4}{r^4},\end{array}$$

where $r$ is the Cartesian Kerr-Schild radius, which is equal to the areal radius for a non-spinning black hole. At the horizon we get

$$\begin{array}{r}{b}_h^2=\frac{B_0^2}{16}.\end{array}$$

Finally, we get

$$\begin{array}{r}{B}_0=4\sqrt{b_h^2}=4\sqrt{{\rho }_h}\sqrt{\frac{b_h^2}{{\rho }_h}},\end{array}$$

where the last equality is useful for comparison to papers that give $b_h^2/{\rho }_h$.

To help with comparing to other codes the following script can be used to compute $b_h^2/{\rho }_h$:

#!/bin/env python
 
import numpy as np
 
# Input parameters
B_0 = 18
r_c = 8
rho_c = 1 / 16
gamma = 4 / 3
mass = 1
 
K = 1 / (gamma * rho_c**(gamma - 1)) * ((gamma - 1) * mass) / (
    (2 * r_c - 3 * mass) * (gamma - 1) - mass)
c_s_c = mass / (2 * r_c - 3 * mass)
c_inf = gamma - 1 + (c_s_c - gamma + 1) * np.sqrt(1. + 3. * c_s_c)
rho_inf = ((gamma - 1) * c_inf / (gamma * K *
                                  (gamma - 1 - c_inf)))**(1. / (gamma - 1.))
rho_h = 1. / 16. * (2.5 - 1.5 * gamma)**(
    (3 * gamma - 5) / (2 * (gamma - 1))) * rho_inf / (c_inf**1.5)
 
print("B_0", B_0)
print("r_c: ", r_c)
print("rho_c", rho_c)
print("b_h^2/rho_h: ", B_0**2 / (16. * rho_h))
print("gamma: ", gamma)

Horizon quantities, \f$\gamma=5/3\f$

The density at the horizon is given by:

$$\begin{array}{r}{\rho }_h\simeq \frac{1}{16}\frac{{\rho }_{\mathrm{\infty }}}{u_h{c}_{s,\mathrm{\infty }}^3},\end{array}$$

which gives [171]

$$\begin{array}{r}{\rho }_h\simeq 0.08\frac{{\rho }_{\mathrm{\infty }}}{c_{s,\mathrm{\infty }}^3}.\end{array}$$

The magnetic field $b^2$ is the same as the $\gamma \ne 5/3$.

Member Function Documentation

get_clone()

auto grmhd::Solutions::BondiMichel::get_clone ( ) const -> std::unique_ptr< evolution::initial_data::InitialData >

overridevirtual

Implements evolution::initial_data::InitialData.

Member Data Documentation

bernoulli_constant_squared_minus_one_

double grmhd::Solutions::BondiMichel::bernoulli_constant_squared_minus_one_

protected

Initial value:

=
      std::numeric_limits<double>::signaling_NaN()

help

constexpr Options::String grmhd::Solutions::BondiMichel::help

staticconstexpr

Initial value:

= {
      "Bondi-Michel solution with a radial magnetic field using \n"
      "the Schwarzschild coordinate system. Quantities prefixed with \n"
      "`sonic` refer to field quantities evaluated at the radius \n"
      "where the fluid speed overtakes the sound speed."}

mass_accretion_rate_over_four_pi_

double grmhd::Solutions::BondiMichel::mass_accretion_rate_over_four_pi_

protected

Initial value:

=
      std::numeric_limits<double>::signaling_NaN()

rest_mass_density_at_infinity_

double grmhd::Solutions::BondiMichel::rest_mass_density_at_infinity_

protected

Initial value:

=
      std::numeric_limits<double>::signaling_NaN()

sonic_fluid_speed_squared_

double grmhd::Solutions::BondiMichel::sonic_fluid_speed_squared_

protected

Initial value:

=
      std::numeric_limits<double>::signaling_NaN()

sonic_sound_speed_squared_

double grmhd::Solutions::BondiMichel::sonic_sound_speed_squared_

protected

Initial value:

=
      std::numeric_limits<double>::signaling_NaN()

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The documentation for this class was generated from the following file:

• src/PointwiseFunctions/AnalyticSolutions/GrMhd/BondiMichel.hpp